40 research outputs found
An equilibrium problem for the limiting eigenvalue distribution of rational Toeplitz matrices
We consider the asymptotic behavior of the eigenvalues of Toeplitz matrices
with rational symbol as the size of the matrix goes to infinity. Our main
result is that the weak limit of the normalized eigenvalue counting measure is
a particular component of the unique solution to a vector equilibrium problem.
Moreover, we show that the other components describe the limiting behavior of
certain generalized eigenvalues. In this way, we generalize the recent results
of Duits and Kuijlaars for banded Toeplitz matrices.Comment: 20 pages, 2 figure
Nonintersecting paths with a staircase initial condition
We consider an ensemble of discrete nonintersecting paths starting from
equidistant points and ending at consecutive integers. Our first result is an
explicit formula for the correlation kernel that allows us to analyze the
process as . In that limit we obtain a new general class of
kernels describing the local correlations close to the equidistant starting
points. As the distance between the starting points goes to infinity, the
correlation kernel converges to that of a single random walker. As the distance
to the starting line increases, however, the local correlations converge to the
sine kernel. Thus, this class interpolates between the sine kernel and an
ensemble of independent particles. We also compute the scaled simultaneous
limit, with both the distance between particles and the distance to the
starting line going to infinity, and obtain a process with number variance
saturation, previously studied by Johansson.Comment: 34 pages, 9 figures; reference added, Theorem 2.1 extended, typos
correcte
The two periodic Aztec diamond and matrix valued orthogonal polynomials
We analyze domino tilings of the two-periodic Aztec diamond by means of
matrix valued orthogonal polynomials that we obtain from a reformulation of the
Aztec diamond as a non-intersecting path model with periodic transition
matrices. In a more general framework we express the correlation kernel for the
underlying determinantal point process as a double contour integral that
contains the reproducing kernel of matrix valued orthogonal polynomials. We use
the Riemann-Hilbert problem to simplify this formula for the case of the
two-periodic Aztec diamond.
In the large size limit we recover the three phases of the model known as
solid, liquid and gas. We describe fine asymptotics for the gas phase and at
the cusp points of the liquid-gas boundary, thereby complementing and extending
results of Chhita and Johansson.Comment: 80 pages, 20 figures; This is an extended version of the paper that
is accepted for publication in the Journal of the EM